Abstract

The propagation properties of coherently combined truncated laser beam arrays with beam distortions through non-Kolmogorov turbulence are studied in detail both analytically and numerically. The analytical expressions for the average intensity and the beam width of coherently combined truncated laser beam arrays with beam distortions propagating through turbulence are derived based on the combination of statistical optics methods and the extended Huygens–Fresnel principle. The effect of beam distortions, such as amplitude modulation and phase fluctuation, is studied by numerical examples. The numerical results reveal that phase fluctuations have significant influence on the spreading of coherently combined truncated laser beam arrays in non-Kolmogorov turbulence, and the effects of the phase fluctuations can be negligible as long as the phase fluctuations are controlled under a certain level, i.e., a>0.05 for the situation considered in the paper. Furthermore, large phase fluctuations can convert the beam distribution rapidly to a Gaussian form, vary the spreading, weaken the optimum truncation effects, and suppress the dependence of spreading on the parameters of the non-Kolmogorov turbulence.

© 2012 Optical Society of America

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    [CrossRef]

2011

H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 1016–1021 (2011).
[CrossRef]

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19, 21163–21173 (2011).
[CrossRef]

J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. 36, 3335–3337 (2011).
[CrossRef]

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104, 207–213 (2011).
[CrossRef]

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103, 971–984 (2011).
[CrossRef]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284, 30–37 (2011).
[CrossRef]

2010

2009

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel–Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26, 387–394 (2009).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

2008

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A 10, 015005 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

2007

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers,” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

2006

T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14, 12022–12027 (2006).
[CrossRef]

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[CrossRef]

2005

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[CrossRef]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

2003

2000

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

1999

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
[CrossRef]

1997

B. Lü and B. Zhang, “Propagation and focusing of laser beams with amplitude modulations and phase fluctuations,” Opt. Commun. 135, 361–368 (1997).
[CrossRef]

1995

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).
[CrossRef]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

1992

1985

1982

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shaped and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Anderegg, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Augst, S. J.

Baker, J. T.

Baykal, Y.

X. Li, X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel–Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26, 387–394 (2009).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A 10, 015005 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Benham, V.

Benjamin, W. S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Cai, Y.

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103, 971–984 (2011).
[CrossRef]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284, 30–37 (2011).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel–Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26, 387–394 (2009).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A 10, 015005 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[CrossRef]

Cang, J.

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Cheng, F.

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284, 30–37 (2011).
[CrossRef]

Cheung, E. C.

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Cui, Z.

Culpepper, M. A.

Dang, A.

Dayton, D.

Ding, D.

Dogariu, A.

Doktorov, E. V.

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shaped and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Epp, P.

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel–Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26, 387–394 (2009).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A 10, 015005 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Fan, T. Y.

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers,” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[CrossRef]

Gonglewski, J.

Goodno, G. D.

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Guo, H.

Hammons, D.

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Hammonsm, D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

He, S.

Higgs, C.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Ho, J. G.

Injeyan, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Ji, X.

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104, 207–213 (2011).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Kansky, J. E.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Kars, S. J.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Komine, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[CrossRef]

Korotkova, O.

Lawrence, R. C.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Li, X.

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104, 207–213 (2011).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Liu, X.

Liu, Z.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Long, J. W. H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Lu, C. A.

Lü, B.

B. Lü and B. Zhang, “Propagation and focusing of laser beams with amplitude modulations and phase fluctuations,” Opt. Commun. 135, 361–368 (1997).
[CrossRef]

Luo, B.

Ma, H.

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Ma, Y.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Manes, K. R.

McClellan, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

McNaught, S. J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Molchan, M. A.

Murphy, D. V.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Nelson, D.

Osmon, C. L.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Ou, B.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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Pilkington, D.

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Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Rice, R. R.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Rothenberg, J.

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shaped and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Sanchez, A.

Sanchez, A. D.

Sermutlu, E.

Shaw, S. E.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Shay, T. M.

Shchepakina, E.

Shirai, T.

Simmons, W. W.

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band-shaped and bandwidth modification,” Phys. Rev. A 26, 12–18 (1982).
[CrossRef]

Sollee, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

Spielbusch, B.

Spring, J.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Tang, H.

Thielen, P.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Vlasov, R. A.

Wang, X.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Ward, B.

Weber, M.

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wickham, M.

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Wolf, E.

Wu, G.

Xu, H.

Xu, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Yu, C. X.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Yu, S.

Zhang, B.

B. Lü and B. Zhang, “Propagation and focusing of laser beams with amplitude modulations and phase fluctuations,” Opt. Commun. 135, 361–368 (1997).
[CrossRef]

Zhao, H.

Zhou, P.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
[CrossRef]

Zhu, S.

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103, 971–984 (2011).
[CrossRef]

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
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Appl. Phys. B

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104, 207–213 (2011).
[CrossRef]

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103, 971–984 (2011).
[CrossRef]

Chin. Phys. Lett.

P. Zhou, X. Wang, Y. Ma, H. Ma, Z. Liu, and X. Xu, “Optimal truncation of element beam in a coherent fiber laser array,” Chin. Phys. Lett. 26, 044206 (2009).
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T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammonsm, H. Injeyan, H. Komine, J. W. H. Long, M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, W. S. Benjamin, and M. Wickham, “Brightness-scaling potential of actively coherently combined solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
[CrossRef]

J. Mod. Opt.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
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J. Opt. A

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A 10, 015005 (2008).
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J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

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B. Lü and B. Zhang, “Propagation and focusing of laser beams with amplitude modulations and phase fluctuations,” Opt. Commun. 135, 361–368 (1997).
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F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284, 30–37 (2011).
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Opt. Eng.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagationthrough non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
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Opt. Lett.

T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003).
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[CrossRef]

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
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[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[CrossRef]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
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Other

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of truncated laser beam array.

Fig. 2.
Fig. 2.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of a and σA.

Fig. 3.
Fig. 3.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of f(λ=1μm,α=3.8,a=0.01,σA=0.3).

Fig. 4.
Fig. 4.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of α.

Fig. 5.
Fig. 5.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of l0.

Fig. 6.
Fig. 6.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of L0.

Fig. 7.
Fig. 7.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of C˜n2.

Fig. 8.
Fig. 8.

Average intensity distributions of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence for different values of a(σA=0.3,f=2,t=1,λ=1μm,α=3.8,z=1km,C˜n2=1×1015m3α).

Fig. 9.
Fig. 9.

RMS beam width of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence.

Fig. 10.
Fig. 10.

RMS beam width of coherently combined truncated laser beam arrays with beam distortions for different values of f in non-Kolmogorov turbulence (z=100km, α=3.8).

Fig. 11.
Fig. 11.

Dependence of the RMS beam width of coherently combined truncated laser beam arrays with beam distortions on a in non-Kolmogorov turbulence (z=30km, α=3.8).

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

H(r⃗)={1,|r⃗|a00,|r⃗|>a0,
u(r⃗,z=0)=m=1N[A(r⃗a⃗m)exp(iϕ(r⃗a⃗m))H(r⃗a⃗m)],
J(r⃗1,r⃗2,z=0)=m=1Nn=1N{H(r⃗1a⃗m)H*(r⃗2a⃗n)×A(r⃗1a⃗m)A(r⃗2a⃗n)exp{i[ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)]}},
J(r⃗1,r⃗2,z=0)=m=1Nn=1N{H(r⃗1a⃗m)H*(r⃗2a⃗n)×A(r⃗1a⃗m)A(r⃗2a⃗n)exp{i[ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)]}}.
A(r⃗1a⃗m)A(r⃗2a⃗n)=I0+σA2(A(r⃗1a⃗m)A0(r⃗1a⃗m))(A(r⃗2a⃗n)A0(r⃗2a⃗n))(A(r⃗1a⃗m)A0(r⃗1a⃗m))2(A(r⃗2a⃗n)A0(r⃗2a⃗n))2,
I0=A0(r⃗1a⃗m)A0(r⃗2a⃗n),
A0(r⃗)=A(r⃗),
σA2=(A(r⃗)A0(r⃗))2.
exp{i[ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)]}=exp{12[ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)]2}=exp{σP2[1ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)ϕ(r⃗1a⃗m)2ϕ(r⃗2a⃗n)2]},
σP2=ϕ2(r⃗).
A(r⃗1a⃗m)A(r⃗2a⃗n)=I0+σA2ρA(r⃗1a⃗m,r⃗2a⃗n),
exp{i[ϕ(r⃗1a⃗m)ϕ(r⃗2a⃗n)]}=exp{σP2[1ρP(r⃗1a⃗m,r⃗2a⃗n)]},
ρA(r⃗1,r⃗2)=exp(|r⃗1r⃗2|2/LA2),
ρP(r⃗1,r⃗2)=exp(|r⃗1r⃗2|2/LP2)(1|r⃗1r⃗2|2/LP2).
J(r⃗1,r⃗2,z=0)=m=1Nn=1N{H(r⃗1a⃗m)H*(r⃗2a⃗n)[I0exp(|r⃗mn|2/a2)+σA2exp{|r⃗mn|2/b2}]},
I0=exp(|r⃗1a⃗m|2+|r⃗2a⃗n|2w02),
H(r⃗)=p=1MApexp(Bp|r⃗|2a02),
J(r⃗1,r⃗2,z=0)=p=1Mq=1Mm=1Nn=1N{ApAq*exp(Bp|r⃗1a⃗m|2+Bq*|r⃗2a⃗n|2a02)×[exp(|r⃗1a⃗m|2+|r⃗2a⃗n|2w02)exp(|r⃗mn|2a2)+σA2exp{|r⃗mn|2b2}]}.
I(r⃗,z)=(k2πz)2dr⃗1dr⃗2J(r⃗1,r⃗2,z=0)exp{ik2z[(r⃗r⃗1)2(r⃗r⃗2)2]}×exp[ψ(r⃗,r⃗1)+ψ*(r⃗,r⃗2)]atmosphere,
exp[ψ(r⃗,r⃗1)+ψ*(r⃗,r⃗2)]atmosphere=exp{4π2k2z010κΦn(κ,α)[1J0(κξ|r⃗1r⃗2|)]dκdξ},
Φn(κ,α)=H(α)C˜n2exp(κ2/κm2)(κ2+κ02)α/2,0κ<,3<α<4,
exp[ψ(r⃗,r⃗1)+ψ*(r⃗,r⃗2)]atmosphere=exp{13π2k2zT|r⃗1r⃗2|2},
T=0κ3Φn(κ)dκ=H(α)C˜n22κm2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ04αα2.
I(r⃗,z)=(k2z)2p=1Mq=1Mm=1Mn=1MApAq*×[exp(E1)C1C2exp(F1r⃗2+G1a⃗mr⃗+H1a⃗nr⃗)+σA2exp(E2)C3C4exp(F2r⃗2+G2a⃗mr⃗+H2a⃗nr⃗)],
C1=Bpa02+1w02+1a2ik2z+δ,C2=Bq*a02+1w02+1a2+ik2z+δβ12C1,C3=Bpa02+1b2ik2z+δ,C4=Bq*a02+1b2+ik2z+δβ22C3,
D1=Bpa02+1w021a2,D2=Bq*a02+1w02+1a2(β1C11),D3=Bpa021b2,D4=Bq*a02+1b2(β2C31),
E1=Bpa⃗m2+Bq*a⃗n2a02+a⃗m2+a⃗n2w02+(a⃗ma⃗n)2a2(D1a⃗m+a⃗n/a2)2C1[(1/a2+D1β1/C1)a⃗m+D2a⃗n]2C2,E2=Bpa⃗m2+Bq*a⃗n2a02+(a⃗ma⃗n)2b2(D3a⃗m+a⃗n/b2)2C3[(1/b2+D3β2/C3)a⃗m+D4a⃗n]2C4,
F1=k24z2[1C1+1C2(β1C11)2],F2=k24z2[1C3+1C4(β2C31)2],
G1=ikz[D1C1+1C2(β1C11)(1a2+D1β1C1)],G2=ikz[D3C3+1C4(β2C31)(1b2+D3β2C3)],
H1=ikz[1a2C1+D2C2(β1C11)],H2=ikz[1b2C3+D4C4(β2C31)].
w2(z)=r⃗2I(r⃗,z)dr⃗I(r⃗,z)dr⃗.
w(z)=p=1Mq=1Mm=1Mn=1MApAq*[Q1mn+σA2Q2mn]p=1Mq=1Mm=1Mn=1MApAq*[Q3mn+σA2Q4mn],
Q1mn=exp(E1)C1C2F12(1+(G1a⃗m+H1a⃗n)24F1)exp((G1a⃗m+H1a⃗n)24F1),
Q2mn=exp(E2)C3C4F22(1+(G2a⃗m+H2a⃗n)24F2)exp((G2a⃗m+H2a⃗n)24F2),
Q3mn=exp(E1)C1C2F1exp((G1a⃗m+H1a⃗n)24F1),
Q4mn=exp(E2)C3C4F2exp((G2a⃗m+H2a⃗n)24F2).

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